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Fractional Langevin equation with anti-periodic boundary conditions
Chaos, Solitons and Fractals 114 (2018) 332–337
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Fractional Langevin equation with anti-periodic boundary conditions Hossein Fazli a,∗, Juan J. Nieto b,c a
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Departamento de Estadística, Análisis Matemático y Optimización, Universidad de Santiago de Compostela, Santiago de Compostela 15782 Spain c Real Academia Galega de Ciencias, Spain b
a r t i c l e
i n f o
Article history: Received 14 May 2017 Revised 13 June 2018 Accepted 10 July 2018
a b s t r a c t In this work, we investigate existence and uniqueness of solutions for nonlinear Langevin equation involving two fractional orders with anti-periodic boundary conditions. Our analysis relies on the coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. An example illustrating our approach is also discussed. © 2018 Elsevier Ltd. All rights reserved.
MSC: 26A33 34A08 34A12 Keywords: Langevin equation Fractional order Anti-periodic boundary value problem Coupled fixed point Mixed monotone operator Existence Uniqueness
1. Introduction Fractional differential equations appear naturally in a number of fields such as physics, geophysics, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, etc. For more details and applications, we refer the reader to the books [1–4] and references [5–9]. Fractional order models are in some cases more accurate than integer-order models as fractional order models allow more degrees of freedom. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes [10]. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. Fractional differential equations are also re-
∗
Corresponding author. E-mail addresses: [email protected] (H. Fazli), [email protected] (J.J. Nieto). https://doi.org/10.1016/j.chaos.2018.07.009 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
garded as an alternative model to nonlinear differential equations [11]. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent work on fractional differential equations, see [12–15]. Anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical processes [16] and have recently received considerable attention. For examples and details of anti-periodic boundary conditions, see [17,18] and the references therein. In this paper, we study the existence and uniqueness of solutions for the following anti-periodic boundary value problem of Langevin equation with two different fractional orders: Dβ (Dα + λ )x(t ) = f (t , x(t )), 0 < t < 1, 0 < α ≤ 1, 1 < β ≤ 2, (1.1) x ( 0 ) + x ( 1 ) = 0, D α x ( 0 ) + D α x ( 1 ) = 0, D 2α x ( 0 ) + D 2α x ( 1 ) = 0, where Dα is the Caputo fractional derivative of order α , f : [0, 1] × R → R is a given continuous function and λ is a real number. Furthermore, D 2α is the sequential fractional derivative presented by Miller and Ross [19]
D α u = Dα u, D kα u = D α D (k−1)α u,
( k = 2, 3, · · · ).
(1.2)
Our interest in studying the problem (1.1) comes from its application as a model for physical systems exhibiting anomalous diffusion, i.e., diffusion process with a non-linear relationship to time, see [20]. The Langevin equation plays a significant role in describing the fluctuation phenomenon in
H. Fazli, J.J. Nieto / Chaos, Solitons and Fractals 114 (2018) 332–337
Brownian motion. In fact, it is well known that in many cases the most convenient way of describing the time evolution of the velocity of the Brownian motion is to use the Langevin equation [21–23]. Also, the Langevin equation has been used extensively in many different application areas, such as modelling the evacuation processes [24], photoelectron counting [25], analyzing the stock market [26], studying the fluid suspensions [27], self organization in complex systems [28], protein dynamics [29], deuteron-cluster dynamics [30], etc. For other applications of the Langevin equation in physical chemistry and electrical engineering, one can refer to [31].
Recently, there are some papers dealing with the existence of solutions for nonlinear Langevin equation for a variety of boundary conditions using various methods (contraction mapping and Krasnoselskii’s fixed point, standard fixed point theorems, variational methods, etc.), see [32–36]. Our main aim is to prove existence and uniqueness of solution for (1.1). This is done in Section 3. Our main tool is Coupled fixed point theorems for mixed monotone mappings which are applied in the appropriate partially ordered sets as well as iterative methods, whose description can be found in [37–39], and which we recall in Section 2. The advantage and importance of this method arises from the fact that it is a constructive method that yields sequences that converge to the unique solution of (1.1).
333
Definition 2.4. We call an element (x, y) ∈ X × X a coupled fixed point of the mapping G if G (x, y ) = x,
and
G (y, x ) = y.
Theorem 2.1. Let (X, ) be a partially ordered set and suppose there exists a metric d on X such that (X, d) is a complete metric space. Let G : X × X → X be a mapping having the mixed monotone property on X. Assume that there exists a k ∈ [0, 1) with
k [d (x, u ) + d (y, v )], 2 for each x u and y v.
d (G (x, y ), G (u, v )) ≤
Suppose either G is continuous or X has the following property: (i) if a non-decreasing sequence {xn } → x, then xn x for all n, (ii) if a non-increasing sequence {yn } → y, then yyn for all n. If there exist x0 , y0 ∈ X such that x0 G (x0 , y0 ) and y0 G (y0 , x0 ), then G has a coupled fixed point (x∗ , y∗ ) ∈ X × X. We define the following partial order on the product space X × X:
2. Preliminaries
(x, y ), (x˜, y˜ ) ∈ X × X, (x, y ) (x˜, y˜ ) ⇔ x x˜, y˜ y.
Here, we recall several known definitions and properties from fractional calculus theory. For details, see [1,12].
Theorem 2.2. In addition to the hypothesis of Theorem 2.1, suppose that for every (x, y ), (x˜, y˜ ) ∈ X × X, there exists an element (u, v) ∈ X × X that is comparable to (x, y) and (x˜, y˜ ), then G has a unique coupled fixed point (x∗ , y∗ ).
Definition 2.1. The Riemann–Liouville integral of order α > 0 for the function x is defined as
Iα x(t ) :=
1
(α )
0
t
(t − τ )α−1 x(τ )dτ ,
0 ≤ t ≤ 1.
provided that the right-hand-side integral exists and is finite. Definition 2.2. The α th Caputo derivative of the function x is defined as
Dα x(t ) := In−α x(n ) (t ) t 1 = (t − τ )−α x(n) (τ )dτ , (n − α ) 0
Theorem 2.3. In addition to the hypothesis of Theorem 2.1, suppose that every pair of elements of X has an upper bound or a lower bound in X. Then x∗ = y∗ . Moreover,
lim G n (x0 , y0 ) = x∗ ,
n→∞
where G n (x0 , y0 ) = G G n−1 (x0 , y0 ), G n−1 (y0 , x0 ) . 3. Existence and uniqueness
0 ≤ t ≤ 1,
where n − 1 < α < n and n ∈ N, provided that the right-hand-side integral exists and is finite. We notice that the Caputo derivative of a constant is zero. Lemma 2.1. Let α , β ≥ 0. If x is continuous, then Iα Iβ x = Iα +β x. Lemma 2.2. Let α ≥ 0. If x is continuous, then Dα Iα x = x. Lemma 2.3. Let n ∈ N+ , α ∈ (n − 1, n ). Then, the general solution of the fractional differential equation Dα x(t ) = 0 is given by
x(t ) = c0 + c1 t + c2 t 2 + · · · + cn−1 t n−1 , where ci ∈ R, i = 1, 2, · · · , n. Let (X, ) be a partially ordered set, i.e. a set X equipped with a relation which is reflexive, asymmetric and transitive. An element x∗ of a partially ordered set X is called an upper bound of a subset A of X if xx∗ for each x ∈ A. A lower bound of A is defined similarly, replacing xx∗ above by x∗ x. Two elements x and y in a partially ordered set are comparable if either xy or yx (or both, in which case x = y). Now we present the coupled fixed point theorems in partially ordered which play main role in our discussion. For details, see [37–39]. Definition 2.3. Let (X, ) be a partially ordered set and G : X × X → X. We say that G has the mixed monotone property if G (x, y ) is monotone non-decreasing in x and is monotone non-increasing in y.
In this section, we intend to give an existence and uniqueness result for the problem (1.1). First we establish the equivalence of this problem and the mixed Fredholm–Volterra integral equation. Then we break our discussion up into two subsections. Two cases of interest are discussed separately in Section 3.1 for the case λ ≥ 0 and in Section 3.2 for the case λ < 0. The reason for discussing them separately is that the definition of coupled lower and upper solution is different in these two cases. Definition 3.1. A function x ∈ C[0, 1] with its Caputo derivative of fractional order existing on [0, 1] is a solution of (1.1) if it satisfies (1.1). Lemma 3.1. x(t) is a solution of the problem (1.1) if and only if it is a solution of the nonlinear mixed Fredholm-Volterra integral equation x(t ) =
t
0
(t − τ )α +β −1 f (τ , x(τ )) dτ − λ (α + β )
0
t
(t − τ )α −1 x ( τ )d τ (α )
(1 − τ )α +β −1 λ 1 (1 − τ )α−1 f (τ , x(τ )) dτ + x ( τ )d τ ( α + β ) 2 0 (α ) 0 1 tα 1 (1 − τ )β −1 + − f (τ , x(τ )) dτ 4(α + 1 ) 2(α + 1 ) (β ) 0 1 (1 − τ )β −α−1 (2 − α ) 1 − α 1 + α α + + t − t α +1 f (τ , x(τ )) dτ . (2 + α ) 4 2 (β − α ) 0 −
1 2
1
(3.1) Proof. Let x(t) be a solution of the problem (1.1). From Lemma 2.2, we have
Dβ (Dα + λ )x(t ) − Iβ f (·, x(· ))(t ) = 0.
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H. Fazli, J.J. Nieto / Chaos, Solitons and Fractals 114 (2018) 332–337
Now applying Lemma 2.3, we deduce
and
(Dα + λ )x(t ) − Iβ f (·, x(· ))(t ) = c1 + c2 t,
y0 (t ) ≥ T1 y0 (t ) − T2 x0 (t ),
or equivalently,
Dα x(t ) + λIα x(· )(t ) − Iα +β f (·, x(· ))(t ) − c1
tα
(α + 1 )
− c2
t α +1
for all t ∈ [0, 1].
= 0.
(α + 2 )
Applying Lemma 2.3 again, the general form of the problem (1.1) can be written as x(t ) = Iα +β f (·, x(· ))(t ) − λIα x(· )(t ) + c0 + c1
tα
(α + 1 )
+ c2
t α +1
(α + 2 )
.
(3.2)
Using the anti-periodic boundary conditions for the problem (1.1), we find that 1 (1 − τ )β −α −1 c2 = −(2 − α ) f (τ , x(τ )) dτ , (β − α ) 0 1 1 (1 − τ )β −1 c1 = − f (τ , x(τ )) dτ 2 0 (β ) 1 (2 − α ) (1 − τ )β −α −1 + f (τ , x(τ )) dτ , 2 (β − α ) 0 1 1 (1 − τ )α +β −1 λ 1 (1 − τ )α−1 c0 = − f (τ , x(τ )) dτ + x (τ ) d τ 2 0 (α + β ) 2 0 (α )
1 β −1 1 1 (1 − τ ) − − f (τ , x(τ )) dτ 2(α + 1 ) 2 0 (β ) (2 − α ) 1 (1 − τ )β −α −1 + f (τ , x(τ )) dτ 2 (β − α ) 0
1 (1 − τ )β −α −1 1 − −(2 − α ) f (τ , x(τ )) dτ . 2(α + 2 ) (β − α ) 0 Substituting the values of c0 , c1 , c2 in (3.2), we obtain the solution (3.1).
On the other hand, it is easy to prove that, if x(t) is a solution of the integral equation (3.1), then x(t) is also a solution of the problem (1.1). 3.1. Investigation in the case λ ≥ 0 In this subsection we consider the case in which λ ≥ 0. For simplicity, we define two operators T1 and T2 as follows
T1 x(t ) :=
(t − τ )α+β −1 f (τ , x(τ )) dτ (α + β ) 0 1 λ (1 − τ )α−1 + x ( τ )d τ 2 0 (α ) 1 1 (1 − τ )β −1 + f (τ , x(τ )) dτ 4(α + 1 ) 0 (β ) (2 − α ) 1 − α 1 + α α + + t (2 + α ) 4 2 1 (1 − τ )β −α−1 × f (τ , x(τ )) dτ . (β − α ) 0
and
T2 x(t ) :=
t
(t − τ )α−1 x ( τ )d τ (α ) 0 1 1 (1 − τ )α +β −1 + f (τ , x(τ )) dτ 2 0 (α + β ) 1 tα (1 − τ )β −1 + f (τ , x(τ )) dτ 2(α + 1 ) 0 (β ) 1 (2 − α ) α+1 (1 − τ )β −α−1 + t f (τ , x(τ )) dτ . (2 + α ) (β − α ) 0
λ
t
Definition 3.2. An element (x0 , y0 ) ∈ C[0, 1] × C[0, 1] is called a coupled lower and upper solution of (1.1) if
x0 (t ) ≤ T1 x0 (t ) − T2 y0 (t ),
To prove the main results, we need the following assumptions: (H1) Let f : [0, 1] × R → R be a function such that f( · , x( · ))(t) ∈ C[0, 1] for every x ∈ C[0, 1], (H2) there exists L > 0 such that 0 ≤ f (t, x ) − f (t, y ) ≤ L(x − y ) for all x, y ∈ R, x ≥ y. Theorem 3.1. With the assumptions (H1)-(H2), if the problem (1.1) has a coupled lower and upper solution and = max{1 , 2 } < 1 where
2L |λ| L + + (α + β + 1 ) (α + 1 ) 2(α + 1 )(β + 1 ) 3 + α (2 − α ) 2L + , 4 (2 + α ) (β − α + 1 ) 2|λ| L L 2 = + + (α + 1 ) (α + β + 1 ) (α + 1 )(β + 1 ) (2 − α ) 2L + , (2 + α ) (β − α + 1 )
1 =
then it has a unique solution in C[0, 1]. Proof. Let C = C ([0, 1], R ) denotes the space of all continuous functions from [0, 1] → R. Then C is a partially ordered set if we define the following order relation in C :
x, y ∈ C , x y
⇐⇒
x(t ) ≤ y(t ), t ∈ [0, 1].
And (C , d ) is a complete metric space with metric d (x, y ) = max0≤t≤1 |x(t ) − y(t )|. Obviously, if {xn } is a monotone nondecreasing sequence in C that converges to x ∈ C and {yn } is a monotone non-increasing sequence in C that converges to y ∈ C , then xn x and yyn , for all n. Also, for any x, y ∈ C , the functions max {x, y} and min {x, y} are the upper and lower bounds of x, y, respectively. Also, C × C is a partially ordered set if we define the following order relation in C × C :
(x1 , y1 ) (x2 , y2 ) ⇐⇒ x1 (t ) ≤ x2 (t ), y2 (t ) ≤ y1 (t ), t ∈ [0, 1]. Furthermore, for every (x1 , y1 ), (x2 , y2 ) ∈ C × C , there exists a (max{x1 , x2 }, min{y1 , y2 } ) ∈ C × C that is comparable to (x1 , y1 ) and (x2 , y2 ). Now we define G : C × C → C by G (x, y )(t ) = T1 x(t ) − T2 y(t ). From assumption (H1), it is easy to see that for any x, y ∈ C , we have G(x, y ) ∈ C . So the operator G is well defined. Now we shall show that G has the mixed monotone property. From the non-decreasing assumption of f with respect to the second variable and using the monotonicity of Riemann–Liouville fractional integral operator, we deduce the operators T1 and T2 are non-decreasing operators. On the other hand, G (x1 , y )(t ) − G (x2 , y )(t ) = T1 x1 (t ) − T1 x2 (t ), and G (x, y1 )(t ) − G (x, y2 )(t ) = −(T2 y1 (t ) − T2 y2 (t ) ). Thus, G (x, y ) is monotone non-decreasing in x and monotone nonincreasing in y. For simplicity, we define the function N f : [0, 1] → R as follows
N f x(t ) := f (·, x(· ))(t ). Now, for x1 , y1 , x2 , y2 ∈ C with x1 x2 , y1 y2 , we have
H. Fazli, J.J. Nieto / Chaos, Solitons and Fractals 114 (2018) 332–337
3.2. Investigation in the case λ < 0
G (x1 , y1 )(t ) − G (x2 , y2 )(t )
t (t − τ )α+β −1 N f x1 ( τ ) − N f x2 ( τ ) d τ ( α + β ) 0 t (t − τ )α−1 −λ [y1 (τ ) − y2 (τ )]dτ (α ) 0 1 1 (1 − τ )α +β −1 − N f y1 ( τ ) − N f y2 ( τ ) d τ 2 0 (α + β ) λ 1 (1 − τ )α−1 + [x1 (τ ) − x2 (τ )]dτ 2 0 (α ) 1 1 (1 − τ )β −1 + N f x1 ( τ ) − N f x2 ( τ ) d τ 4(α + 1 ) 0 (β ) 1 tα (1 − τ )β −1 − N f y1 ( τ ) − N f y2 ( τ ) 2(α + 1 ) 0 (β ) (2 − α ) 1 − α 1 + α α 1 (1 − τ )β −α−1 + + t (2 + α ) 4 2 (β − α ) 0 × N f x1 (τ ) − N f x2 (τ ) dτ .
To derive the existence and uniqueness result in this case, we can apply the same technique as in previous subsection. The main difference is the structure of the function G and consequently the definition of coupled lower and upper solution of the problem (1.1). We could carry out a similar argument to prove existence and uniqueness result. We do not try to give here an account of the extremely wide details on this topic, we only confine ourself to state the conditions and the definition of coupled lower and upper solution, and omit the full details of the processes. Similarly, we first define two operators T3 and T4 as follows
=
−
(2 − α ) α+1 t (2 + α )
1
(1 − τ )β −α−1
T3 x(t ) :=
N f y 1 ( τ ) − N f y 2 ( τ ) d τ .
2
tα
(β − α )
0
f (τ , x(τ )) dτ .
T4 x(t ) := −
λ 2
1
0
(1 − τ )α −1 1 x ( τ )d τ + (α ) 2
tα + 2(α + 1 )
1
0
(2 − α ) α +1 + t (2 + α )
1
0
(1 − τ )α +β −1 f (τ , x(τ )) dτ (α + β )
(1 − τ )β −1 f (τ , x(τ )) dτ (β )
0
1
(1 − τ )β −α −1 f (τ , x(τ )) dτ . (β − α )
Definition 3.3. An element (x0 , y0 ) ∈ C × C is called a coupled lower and upper solution of (1.1) if
(H1) Let f : [0, 1] × R → R be a function such that f (·, x(· ))(t ) ∈ C for every x ∈ C , (H2) there exists L > 0 such that 0 ≤ f (t, x ) − f (t, y ) ≤ L(x − y ) for all x, y ∈ R, x ≥ y. Theorem 3.2. With the assumptions (H1) and (H2), if the problem (1.1) has a coupled lower and upper solution and = max{1 , 2 } < 1 where
Remark 3.1. In view of Theorem 3.2, the sequences {xn } and {yn } defined by
n = 1, 2, · · · , (3.3)
where x0 and y0 are the coupled lower and upper solutions of problem (1.1), generate a sequence {(xn , yn )} which each of its elements is a coupled lower and upper solution of problem (1.1). Remark 3.2. Let (x0 , y0 ) be a coupled lower and upper solution of problem (1.1) such that x0 y0 and let {xn } and {yn } be the sequences defined by (3.3). Then
x0 x1 · · · xn · · · x · · · yn · · · y1 y0 , and both sequences converge to x∗ in C[0, 1].
+
4
To prove the main results, we need the following assumptions:
Thus an application of the Theorems 2.2 and 2.3, together with Lemma 3.1, yields the existence and uniqueness of the solution of x∗ ∈ C of the problem (1.1). Moreover, the unique solution of n n (1.1) can be obtained as lim n→∞ G (x0 , y0 ) where G (x0 , y0 ) = G G n−1 (x0 , y0 ), G n−1 (y0 , x0 ) .
∗
(2 + α )
for all t ∈ [0, 1].
(d (x1 , x2 ) + d (y1 , y2 ) ).
yn = G (yn−1 , xn−1 ),
0
1
y0 (t ) ≥ T3 y0 (t ) − T4 x0 (t ),
L
xn = G (xn−1 , yn−1 ),
(t − τ )α −1 x ( τ )d τ (α )
t
and
1 1 1 d (x1 , x2 ) + 2 d (y1 , y2 ) 2 2 2
x0 (t ) ≤ T3 x0 (t ) − T4 y0 (t ),
d (x , x ) (β − α + 1 ) 1 2 (2 − α ) L + d (y , y ) (2 + α ) (β − α + 1 ) 1 2
(t − τ )α +β −1 f (τ , x(τ )) dτ − λ (α + β )
and
(2 − α ) 1 − α 1 + α α + max | + t 0≤t≤1 (2 + α ) 4 2
≤
0
+
L d ( x1 , x2 ) 4(α + 1 )(β + 1 ) L + d ( y1 , y2 ) 2(α + 1 )(β + 1 )
=
t
(1 − τ )β −1 + f (τ , x(τ )) dτ 4(α + 1 ) 0 (β ) (2 − α ) 1 − α 1 + α 1 (1 − τ )β −α −1
+
×
1
(β − α ) L |λ| ≤ d (x , x ) + d (y , y ) (α + β + 1 ) 1 2 (α + 1 ) 1 2 L |λ| + d ( y1 , y2 ) + d ( x1 , x2 ) 2(α + β + 1 ) 2(α + 1 ) 0
335
2L 2|λ| L + + (α + β + 1 ) (α + 1 ) 2(α + 1 )(β + 1 ) 3 + α (2 − α ) 2L + , 4 (2 + α ) (β − α + 1 ) |λ| L L 2 = + + (α + 1 ) (α + β + 1 ) (α + 1 )(β + 1 ) (2 − α ) 2L + , (2 + α ) (β − α + 1 )
1 =
then it has a unique solution in C . Proof. It suffices to define G : C × C → C by G (x, y )(t ) = T3 x(t ) − T4 y(t ), and to apply Theorem 3.2.
the
same
arguments
as
in
the
proof
Example 3.1. Let us consider the following problem
3 1 1 D 2 D 2 + 14 x(t ) = 10 (x(t ) cos t ) − 1, 0 < t < 1, 1
1
x ( 0 ) + x ( 1 ) = 0, D 2 x ( 0 ) + D 2 x ( 1 ) = 0, D
21 2
x (0 ) + D
21 2
x ( 1 ) = 0.
of
(3.4)
336
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1 1 3 1 10 (x (t ) cos t ) − 1, α = 2 , β = 2 and λ = 4 . Observe that, for every x ≥ y, we have 0 ≤ f (t, x ) − f (t, y ) ≤ 1 1 10 (x − y ) with L = 10 and
Here f (t, x ) =
2L |λ| L + + (α + β + 1 ) (α + 1 ) 2(α + 1 )(β + 1 ) 3 + α (2 − α ) 2L + = 0.42644, 4 (2 + α ) (β − α + 1 ) 2|λ| L L 2 = + + (α + 1 ) (α + β + 1 ) (α + 1 )(β + 1 ) (2 − α ) 2L + = 0.65289, (2 + α ) (β − α + 1 )
1 =
and = max{1 , 2 } = 0.65289 < 1. Now from (3.3) we let G : C × C → C by G (x, y )(t ) =
t
(t − τ )α+β −1 1
(x(τ ) cos τ ) − 1 (α + β ) 10 t (t − τ )α−1 ×dτ − λ y ( τ )d τ (α ) 0 1 1 (1 − τ )α+β −1 1 − (y(τ ) cos τ ) − 1 dτ 2 0 (α + β ) 10 1 λ (1 − τ )α−1 + x ( τ )d τ 2 0 (α ) 1 1 (1 − τ )β −1 1 + (x(τ ) cos τ ) − 1 dτ 4(α + 1 ) 0 (β ) 10 1 α β −1 t (1 − τ ) 1 − (y(τ ) cos τ ) − 1 2(α + 1 ) 0 (β ) 10 (2 − α ) 1 − α 1 + α α 1 (1 − τ )β −α−1 + + t (2 + α ) 4 2 (β − α ) 0 1 × (x(τ ) cos τ ) − 1 dτ
fractional orders in different intervals. As a first step, we establish the equivalence of this problem and the corresponding integral equation by applying the tools of fractional calculus and define a fixed point problem. The existence and uniqueness of solutions of these equations are obtained using Coupled fixed point theorems for mixed monotone mappings. Our approach is simple and is applicable to a variety of real world problems. For the illustration of the results, we have considered an example. As a special case, the existence results for a nonlinear thirdorder differential equations with anti-periodic boundary conditions:
D3 + λD2 x(t ) = f (t , x(t ) ),
x(0 ) + x(1 ) = 0, x (0 ) + x (1 ) = 0, x (0 ) + x (1 ) = 0, can be obtained by fixing α = 1 and β = 2 in the results of this paper.
0
10
(2 − α ) α+1 1 (1 − τ )β −α−1 − t (2 + α ) (β − α ) 0 1 × (y(τ ) cos τ ) − 1 dτ . 10
A relatively simple calculus, with the help of Maple, shows that (x0 (t ), y0 (t )) = (−1, 1 ) is a coupled lower and upper solution of problem (3.4). Therefore, all the assumption of Theorem 3.2 hold and consequently, problem (3.4) has a unique solution in C[0, 1]. Moreover, the unique solution of n n (3.4) can be obtained as lim n→∞ G (x0 , y0 ) where G (x0 , y0 ) = G G n−1 (x0 , y0 ), G n−1 (y0 , x0 ) . Using simple calculus, with the help of Maple, we have G (x0 , y0 ) = −0.33544 − 0.50 0 0 0t 2 + 0.10 0 0 0 cos(t )
√ 3 −0.43755 t + 0.61057t 2 ,
G (y0 , x0 ) = 0.24440 − 0.50 0 0 0t 2 − 0.10 0 0 0 cos(t )
√ 3 +0.28631 t + 0.72274t 2 .
It is interesting to point out that (G n (x0 , y0 ), G n (y0 , x0 ) ) n = 1, 2, 3, · · · serve as an approximation to the unique coupled fixed point of G of increasing accuracy as n → ∞. On the other hand, from Theorem 2.3, the unique solution of (3.4) can be obtained as
x∗ = lim G n (x0 , y0 ) = lim G n (y0 , x0 ). n→∞
n→∞
4. Conclusions In this paper, we have presented some results dealing with the existence and uniqueness of solutions for an anti-periodic boundary value problem of nonlinear Langevin equation involving two
Acknowledgments The first author would like to thank University of Tabriz for the financial support of this research. The work of J.J. Nieto has been partially supported by the AEI of Spain under Grant MTM201675140-P and co-financed by European Community fund FEDER, and XUNTA de Galicia under grants GRC2015-004 and R2016/022. References [1] Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999. [2] Sabatier J, Agrawal OP, Machado JAT, editors. Advances in fractional calculus: theoretical developments and applications in physics and engineering. Dordrecht: Springer; 2007. [3] Zaslavsky GM. Hamiltonian chaos and fractional dynamics. Oxford: Oxford University Press; 2005. [4] Magin RL. Fractional calculus in bioengineering. Connecticut: Begell House Publisher, Inc.; 2006. [5] Abdeljawad T. On Riemann and Caputo fractional differences. Comput Math Appl 2011;62:1602–11. [6] Abdeljawad T, Baleanu D. Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function. Commun Nonlinear Sci Numer Simul 2011;16:4682–8. [7] Fazli H, Bahrami F. On the steady solutions of fractional reaction-diffusion equations. Filomat 2017;31(6):1655–64. [8] Bahrami F, Fazli H, Akbarfam AJ. A new approach on fractional variational problems and euler-lagrange equations. Commun Nonlinear Sci Numer Simul 2015;23(13):39–50. [9] Sadallah M, Muslih SI, Baleanu D, Rabei E. Fractional time action and perturbed gravity. Fractals 2011;19:243–7. [10] Lazarevic MP, Spasic AM. Finite-time stability analysis of fractional order time delay systems: Gronwall’s approach. Math Comput Model 2009;49:475–81. [11] Bonilla B, Rivero M, Rodriguez-Germa L, Trujillo JJ. Fractional differential equations as alternative models to nonlinear differential equations. Appl Math Comput 2007;187:79–88. [12] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. North-Holland mathematics studies, vol. 204. Amsterdam: Elsevier Science B.V.; 2006. [13] Ahmad B, Nieto JJ. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput Math Appl 2009;58:1838–43. [14] Fazli H, Nieto JJ, Bahrami F. On the existence and uniqueness results for nonlinear sequential fractional differential equations. Appl Comput Math 2018;17(1):36–47. [15] Baleanu D, Mustafa OG, Agarwal RP. An existence result for a superlinear fractional differential equation. Appl Math Lett 2010;23:1129–32. [16] Negele JW, Vogt E, editors. Volume 23 of advances in the physics of particles and nuclei, Advances in nuclear physics. Springer Science and Business Media; 1996. [17] Ahmad B. Existence of solutions for fractional differential equations of order q ∈ (2, 3] with anti-periodic boundary conditions. J Appl Math Comput 2010;34:385–91. [18] Chen Y, Nieto JJ, O’Regan D. Anti-periodic solutions for evolution equations associated with maximal monotone mappings. Appl Math Lett 2011;24(3):302–7. [19] Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. New York: Wiley and Sons; 1993. [20] Porra JM, Wang KG, Masoliver J. Generalized Langevin equations: anomalous diffusion and probability distributions. Phys Rev E 1996;53:5872–81. [21] Wax N., (Ed.) Selected papers on noise and stochastic processes. New York: Dover; 1954.
H. Fazli, J.J. Nieto / Chaos, Solitons and Fractals 114 (2018) 332–337 [22] Mazo R. Brownian motion: fluctuations. Dynamics and applications. Oxford: Oxford Univ. Press; 2002. [23] Zwanzig R. Nonequilibrium statistical mechanics. New York: Oxford University Press; 2001. [24] Kosinski RA, Grabowski A. Langevin equations for modeling evacuation processes. Acta Phys Pol B 2010;3(2):365–77. [25] Wodkiewicz K, Zubairy MS. Exact solution of a nonlinear Langevin equation with applications to photoelectron counting and noise-induced instability. J Math Phys 1983;24(6):1401–4. [26] Bouchaud JP, Cont R. A Langevin approach to stock market fluctuations and crashes. Eur Phys J B 1998;6(4):543–50. [27] Hinch EJ. Application of the Langevin equation to fluid suspensions. J Fluid Mech 1975;72(3):499–511. [28] Fraaije JGEM, Zvelindovsky AV, Sevink GJA, Maurits NM. Modulated self-organization in complex amphilic systems. Mol Simul 20 0 0;25(3–4):131–44. [29] Schluttig J, Alamanova D, Helms V, Schwarz US. Dynamics of protein-protein encounter: a Langevin equation approach with reaction patches. J Chem Phys 2008;129(15):155106. [30] Takahashi A. Low-energy nuclear reactions and new energy technologies sourcebook. Oxford University Press; 2009. [31] Coffey WT, Kalmykov YP, Waldron JT. The Langevin equation. Second edition. Singapore: World Scientific; 2004.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Fractional Langevin equation with anti-periodic boundary conditions Hossein Fazli a,∗, Juan J. Nieto b,c a
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Departamento de Estadística, Análisis Matemático y Optimización, Universidad de Santiago de Compostela, Santiago de Compostela 15782 Spain c Real Academia Galega de Ciencias, Spain b
a r t i c l e
i n f o
Article history: Received 14 May 2017 Revised 13 June 2018 Accepted 10 July 2018
a b s t r a c t In this work, we investigate existence and uniqueness of solutions for nonlinear Langevin equation involving two fractional orders with anti-periodic boundary conditions. Our analysis relies on the coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. An example illustrating our approach is also discussed. © 2018 Elsevier Ltd. All rights reserved.
MSC: 26A33 34A08 34A12 Keywords: Langevin equation Fractional order Anti-periodic boundary value problem Coupled fixed point Mixed monotone operator Existence Uniqueness
1. Introduction Fractional differential equations appear naturally in a number of fields such as physics, geophysics, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, etc. For more details and applications, we refer the reader to the books [1–4] and references [5–9]. Fractional order models are in some cases more accurate than integer-order models as fractional order models allow more degrees of freedom. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes [10]. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. Fractional differential equations are also re-
∗
Corresponding author. E-mail addresses: [email protected] (H. Fazli), [email protected] (J.J. Nieto). https://doi.org/10.1016/j.chaos.2018.07.009 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
garded as an alternative model to nonlinear differential equations [11]. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent work on fractional differential equations, see [12–15]. Anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical processes [16] and have recently received considerable attention. For examples and details of anti-periodic boundary conditions, see [17,18] and the references therein. In this paper, we study the existence and uniqueness of solutions for the following anti-periodic boundary value problem of Langevin equation with two different fractional orders: Dβ (Dα + λ )x(t ) = f (t , x(t )), 0 < t < 1, 0 < α ≤ 1, 1 < β ≤ 2, (1.1) x ( 0 ) + x ( 1 ) = 0, D α x ( 0 ) + D α x ( 1 ) = 0, D 2α x ( 0 ) + D 2α x ( 1 ) = 0, where Dα is the Caputo fractional derivative of order α , f : [0, 1] × R → R is a given continuous function and λ is a real number. Furthermore, D 2α is the sequential fractional derivative presented by Miller and Ross [19]
D α u = Dα u, D kα u = D α D (k−1)α u,
( k = 2, 3, · · · ).
(1.2)
Our interest in studying the problem (1.1) comes from its application as a model for physical systems exhibiting anomalous diffusion, i.e., diffusion process with a non-linear relationship to time, see [20]. The Langevin equation plays a significant role in describing the fluctuation phenomenon in
H. Fazli, J.J. Nieto / Chaos, Solitons and Fractals 114 (2018) 332–337
Brownian motion. In fact, it is well known that in many cases the most convenient way of describing the time evolution of the velocity of the Brownian motion is to use the Langevin equation [21–23]. Also, the Langevin equation has been used extensively in many different application areas, such as modelling the evacuation processes [24], photoelectron counting [25], analyzing the stock market [26], studying the fluid suspensions [27], self organization in complex systems [28], protein dynamics [29], deuteron-cluster dynamics [30], etc. For other applications of the Langevin equation in physical chemistry and electrical engineering, one can refer to [31].
Recently, there are some papers dealing with the existence of solutions for nonlinear Langevin equation for a variety of boundary conditions using various methods (contraction mapping and Krasnoselskii’s fixed point, standard fixed point theorems, variational methods, etc.), see [32–36]. Our main aim is to prove existence and uniqueness of solution for (1.1). This is done in Section 3. Our main tool is Coupled fixed point theorems for mixed monotone mappings which are applied in the appropriate partially ordered sets as well as iterative methods, whose description can be found in [37–39], and which we recall in Section 2. The advantage and importance of this method arises from the fact that it is a constructive method that yields sequences that converge to the unique solution of (1.1).
333
Definition 2.4. We call an element (x, y) ∈ X × X a coupled fixed point of the mapping G if G (x, y ) = x,
and
G (y, x ) = y.
Theorem 2.1. Let (X, ) be a partially ordered set and suppose there exists a metric d on X such that (X, d) is a complete metric space. Let G : X × X → X be a mapping having the mixed monotone property on X. Assume that there exists a k ∈ [0, 1) with
k [d (x, u ) + d (y, v )], 2 for each x u and y v.
d (G (x, y ), G (u, v )) ≤
Suppose either G is continuous or X has the following property: (i) if a non-decreasing sequence {xn } → x, then xn x for all n, (ii) if a non-increasing sequence {yn } → y, then yyn for all n. If there exist x0 , y0 ∈ X such that x0 G (x0 , y0 ) and y0 G (y0 , x0 ), then G has a coupled fixed point (x∗ , y∗ ) ∈ X × X. We define the following partial order on the product space X × X:
2. Preliminaries
(x, y ), (x˜, y˜ ) ∈ X × X, (x, y ) (x˜, y˜ ) ⇔ x x˜, y˜ y.
Here, we recall several known definitions and properties from fractional calculus theory. For details, see [1,12].
Theorem 2.2. In addition to the hypothesis of Theorem 2.1, suppose that for every (x, y ), (x˜, y˜ ) ∈ X × X, there exists an element (u, v) ∈ X × X that is comparable to (x, y) and (x˜, y˜ ), then G has a unique coupled fixed point (x∗ , y∗ ).
Definition 2.1. The Riemann–Liouville integral of order α > 0 for the function x is defined as
Iα x(t ) :=
1
(α )
0
t
(t − τ )α−1 x(τ )dτ ,
0 ≤ t ≤ 1.
provided that the right-hand-side integral exists and is finite. Definition 2.2. The α th Caputo derivative of the function x is defined as
Dα x(t ) := In−α x(n ) (t ) t 1 = (t − τ )−α x(n) (τ )dτ , (n − α ) 0
Theorem 2.3. In addition to the hypothesis of Theorem 2.1, suppose that every pair of elements of X has an upper bound or a lower bound in X. Then x∗ = y∗ . Moreover,
lim G n (x0 , y0 ) = x∗ ,
n→∞
where G n (x0 , y0 ) = G G n−1 (x0 , y0 ), G n−1 (y0 , x0 ) . 3. Existence and uniqueness
0 ≤ t ≤ 1,
where n − 1 < α < n and n ∈ N, provided that the right-hand-side integral exists and is finite. We notice that the Caputo derivative of a constant is zero. Lemma 2.1. Let α , β ≥ 0. If x is continuous, then Iα Iβ x = Iα +β x. Lemma 2.2. Let α ≥ 0. If x is continuous, then Dα Iα x = x. Lemma 2.3. Let n ∈ N+ , α ∈ (n − 1, n ). Then, the general solution of the fractional differential equation Dα x(t ) = 0 is given by
x(t ) = c0 + c1 t + c2 t 2 + · · · + cn−1 t n−1 , where ci ∈ R, i = 1, 2, · · · , n. Let (X, ) be a partially ordered set, i.e. a set X equipped with a relation which is reflexive, asymmetric and transitive. An element x∗ of a partially ordered set X is called an upper bound of a subset A of X if xx∗ for each x ∈ A. A lower bound of A is defined similarly, replacing xx∗ above by x∗ x. Two elements x and y in a partially ordered set are comparable if either xy or yx (or both, in which case x = y). Now we present the coupled fixed point theorems in partially ordered which play main role in our discussion. For details, see [37–39]. Definition 2.3. Let (X, ) be a partially ordered set and G : X × X → X. We say that G has the mixed monotone property if G (x, y ) is monotone non-decreasing in x and is monotone non-increasing in y.
In this section, we intend to give an existence and uniqueness result for the problem (1.1). First we establish the equivalence of this problem and the mixed Fredholm–Volterra integral equation. Then we break our discussion up into two subsections. Two cases of interest are discussed separately in Section 3.1 for the case λ ≥ 0 and in Section 3.2 for the case λ < 0. The reason for discussing them separately is that the definition of coupled lower and upper solution is different in these two cases. Definition 3.1. A function x ∈ C[0, 1] with its Caputo derivative of fractional order existing on [0, 1] is a solution of (1.1) if it satisfies (1.1). Lemma 3.1. x(t) is a solution of the problem (1.1) if and only if it is a solution of the nonlinear mixed Fredholm-Volterra integral equation x(t ) =
t
0
(t − τ )α +β −1 f (τ , x(τ )) dτ − λ (α + β )
0
t
(t − τ )α −1 x ( τ )d τ (α )
(1 − τ )α +β −1 λ 1 (1 − τ )α−1 f (τ , x(τ )) dτ + x ( τ )d τ ( α + β ) 2 0 (α ) 0 1 tα 1 (1 − τ )β −1 + − f (τ , x(τ )) dτ 4(α + 1 ) 2(α + 1 ) (β ) 0 1 (1 − τ )β −α−1 (2 − α ) 1 − α 1 + α α + + t − t α +1 f (τ , x(τ )) dτ . (2 + α ) 4 2 (β − α ) 0 −
1 2
1
(3.1) Proof. Let x(t) be a solution of the problem (1.1). From Lemma 2.2, we have
Dβ (Dα + λ )x(t ) − Iβ f (·, x(· ))(t ) = 0.
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Now applying Lemma 2.3, we deduce
and
(Dα + λ )x(t ) − Iβ f (·, x(· ))(t ) = c1 + c2 t,
y0 (t ) ≥ T1 y0 (t ) − T2 x0 (t ),
or equivalently,
Dα x(t ) + λIα x(· )(t ) − Iα +β f (·, x(· ))(t ) − c1
tα
(α + 1 )
− c2
t α +1
for all t ∈ [0, 1].
= 0.
(α + 2 )
Applying Lemma 2.3 again, the general form of the problem (1.1) can be written as x(t ) = Iα +β f (·, x(· ))(t ) − λIα x(· )(t ) + c0 + c1
tα
(α + 1 )
+ c2
t α +1
(α + 2 )
.
(3.2)
Using the anti-periodic boundary conditions for the problem (1.1), we find that 1 (1 − τ )β −α −1 c2 = −(2 − α ) f (τ , x(τ )) dτ , (β − α ) 0 1 1 (1 − τ )β −1 c1 = − f (τ , x(τ )) dτ 2 0 (β ) 1 (2 − α ) (1 − τ )β −α −1 + f (τ , x(τ )) dτ , 2 (β − α ) 0 1 1 (1 − τ )α +β −1 λ 1 (1 − τ )α−1 c0 = − f (τ , x(τ )) dτ + x (τ ) d τ 2 0 (α + β ) 2 0 (α )
1 β −1 1 1 (1 − τ ) − − f (τ , x(τ )) dτ 2(α + 1 ) 2 0 (β ) (2 − α ) 1 (1 − τ )β −α −1 + f (τ , x(τ )) dτ 2 (β − α ) 0
1 (1 − τ )β −α −1 1 − −(2 − α ) f (τ , x(τ )) dτ . 2(α + 2 ) (β − α ) 0 Substituting the values of c0 , c1 , c2 in (3.2), we obtain the solution (3.1).
On the other hand, it is easy to prove that, if x(t) is a solution of the integral equation (3.1), then x(t) is also a solution of the problem (1.1). 3.1. Investigation in the case λ ≥ 0 In this subsection we consider the case in which λ ≥ 0. For simplicity, we define two operators T1 and T2 as follows
T1 x(t ) :=
(t − τ )α+β −1 f (τ , x(τ )) dτ (α + β ) 0 1 λ (1 − τ )α−1 + x ( τ )d τ 2 0 (α ) 1 1 (1 − τ )β −1 + f (τ , x(τ )) dτ 4(α + 1 ) 0 (β ) (2 − α ) 1 − α 1 + α α + + t (2 + α ) 4 2 1 (1 − τ )β −α−1 × f (τ , x(τ )) dτ . (β − α ) 0
and
T2 x(t ) :=
t
(t − τ )α−1 x ( τ )d τ (α ) 0 1 1 (1 − τ )α +β −1 + f (τ , x(τ )) dτ 2 0 (α + β ) 1 tα (1 − τ )β −1 + f (τ , x(τ )) dτ 2(α + 1 ) 0 (β ) 1 (2 − α ) α+1 (1 − τ )β −α−1 + t f (τ , x(τ )) dτ . (2 + α ) (β − α ) 0
λ
t
Definition 3.2. An element (x0 , y0 ) ∈ C[0, 1] × C[0, 1] is called a coupled lower and upper solution of (1.1) if
x0 (t ) ≤ T1 x0 (t ) − T2 y0 (t ),
To prove the main results, we need the following assumptions: (H1) Let f : [0, 1] × R → R be a function such that f( · , x( · ))(t) ∈ C[0, 1] for every x ∈ C[0, 1], (H2) there exists L > 0 such that 0 ≤ f (t, x ) − f (t, y ) ≤ L(x − y ) for all x, y ∈ R, x ≥ y. Theorem 3.1. With the assumptions (H1)-(H2), if the problem (1.1) has a coupled lower and upper solution and = max{1 , 2 } < 1 where
2L |λ| L + + (α + β + 1 ) (α + 1 ) 2(α + 1 )(β + 1 ) 3 + α (2 − α ) 2L + , 4 (2 + α ) (β − α + 1 ) 2|λ| L L 2 = + + (α + 1 ) (α + β + 1 ) (α + 1 )(β + 1 ) (2 − α ) 2L + , (2 + α ) (β − α + 1 )
1 =
then it has a unique solution in C[0, 1]. Proof. Let C = C ([0, 1], R ) denotes the space of all continuous functions from [0, 1] → R. Then C is a partially ordered set if we define the following order relation in C :
x, y ∈ C , x y
⇐⇒
x(t ) ≤ y(t ), t ∈ [0, 1].
And (C , d ) is a complete metric space with metric d (x, y ) = max0≤t≤1 |x(t ) − y(t )|. Obviously, if {xn } is a monotone nondecreasing sequence in C that converges to x ∈ C and {yn } is a monotone non-increasing sequence in C that converges to y ∈ C , then xn x and yyn , for all n. Also, for any x, y ∈ C , the functions max {x, y} and min {x, y} are the upper and lower bounds of x, y, respectively. Also, C × C is a partially ordered set if we define the following order relation in C × C :
(x1 , y1 ) (x2 , y2 ) ⇐⇒ x1 (t ) ≤ x2 (t ), y2 (t ) ≤ y1 (t ), t ∈ [0, 1]. Furthermore, for every (x1 , y1 ), (x2 , y2 ) ∈ C × C , there exists a (max{x1 , x2 }, min{y1 , y2 } ) ∈ C × C that is comparable to (x1 , y1 ) and (x2 , y2 ). Now we define G : C × C → C by G (x, y )(t ) = T1 x(t ) − T2 y(t ). From assumption (H1), it is easy to see that for any x, y ∈ C , we have G(x, y ) ∈ C . So the operator G is well defined. Now we shall show that G has the mixed monotone property. From the non-decreasing assumption of f with respect to the second variable and using the monotonicity of Riemann–Liouville fractional integral operator, we deduce the operators T1 and T2 are non-decreasing operators. On the other hand, G (x1 , y )(t ) − G (x2 , y )(t ) = T1 x1 (t ) − T1 x2 (t ), and G (x, y1 )(t ) − G (x, y2 )(t ) = −(T2 y1 (t ) − T2 y2 (t ) ). Thus, G (x, y ) is monotone non-decreasing in x and monotone nonincreasing in y. For simplicity, we define the function N f : [0, 1] → R as follows
N f x(t ) := f (·, x(· ))(t ). Now, for x1 , y1 , x2 , y2 ∈ C with x1 x2 , y1 y2 , we have
H. Fazli, J.J. Nieto / Chaos, Solitons and Fractals 114 (2018) 332–337
3.2. Investigation in the case λ < 0
G (x1 , y1 )(t ) − G (x2 , y2 )(t )
t (t − τ )α+β −1 N f x1 ( τ ) − N f x2 ( τ ) d τ ( α + β ) 0 t (t − τ )α−1 −λ [y1 (τ ) − y2 (τ )]dτ (α ) 0 1 1 (1 − τ )α +β −1 − N f y1 ( τ ) − N f y2 ( τ ) d τ 2 0 (α + β ) λ 1 (1 − τ )α−1 + [x1 (τ ) − x2 (τ )]dτ 2 0 (α ) 1 1 (1 − τ )β −1 + N f x1 ( τ ) − N f x2 ( τ ) d τ 4(α + 1 ) 0 (β ) 1 tα (1 − τ )β −1 − N f y1 ( τ ) − N f y2 ( τ ) 2(α + 1 ) 0 (β ) (2 − α ) 1 − α 1 + α α 1 (1 − τ )β −α−1 + + t (2 + α ) 4 2 (β − α ) 0 × N f x1 (τ ) − N f x2 (τ ) dτ .
To derive the existence and uniqueness result in this case, we can apply the same technique as in previous subsection. The main difference is the structure of the function G and consequently the definition of coupled lower and upper solution of the problem (1.1). We could carry out a similar argument to prove existence and uniqueness result. We do not try to give here an account of the extremely wide details on this topic, we only confine ourself to state the conditions and the definition of coupled lower and upper solution, and omit the full details of the processes. Similarly, we first define two operators T3 and T4 as follows
=
−
(2 − α ) α+1 t (2 + α )
1
(1 − τ )β −α−1
T3 x(t ) :=
N f y 1 ( τ ) − N f y 2 ( τ ) d τ .
2
tα
(β − α )
0
f (τ , x(τ )) dτ .
T4 x(t ) := −
λ 2
1
0
(1 − τ )α −1 1 x ( τ )d τ + (α ) 2
tα + 2(α + 1 )
1
0
(2 − α ) α +1 + t (2 + α )
1
0
(1 − τ )α +β −1 f (τ , x(τ )) dτ (α + β )
(1 − τ )β −1 f (τ , x(τ )) dτ (β )
0
1
(1 − τ )β −α −1 f (τ , x(τ )) dτ . (β − α )
Definition 3.3. An element (x0 , y0 ) ∈ C × C is called a coupled lower and upper solution of (1.1) if
(H1) Let f : [0, 1] × R → R be a function such that f (·, x(· ))(t ) ∈ C for every x ∈ C , (H2) there exists L > 0 such that 0 ≤ f (t, x ) − f (t, y ) ≤ L(x − y ) for all x, y ∈ R, x ≥ y. Theorem 3.2. With the assumptions (H1) and (H2), if the problem (1.1) has a coupled lower and upper solution and = max{1 , 2 } < 1 where
Remark 3.1. In view of Theorem 3.2, the sequences {xn } and {yn } defined by
n = 1, 2, · · · , (3.3)
where x0 and y0 are the coupled lower and upper solutions of problem (1.1), generate a sequence {(xn , yn )} which each of its elements is a coupled lower and upper solution of problem (1.1). Remark 3.2. Let (x0 , y0 ) be a coupled lower and upper solution of problem (1.1) such that x0 y0 and let {xn } and {yn } be the sequences defined by (3.3). Then
x0 x1 · · · xn · · · x · · · yn · · · y1 y0 , and both sequences converge to x∗ in C[0, 1].
+
4
To prove the main results, we need the following assumptions:
Thus an application of the Theorems 2.2 and 2.3, together with Lemma 3.1, yields the existence and uniqueness of the solution of x∗ ∈ C of the problem (1.1). Moreover, the unique solution of n n (1.1) can be obtained as lim n→∞ G (x0 , y0 ) where G (x0 , y0 ) = G G n−1 (x0 , y0 ), G n−1 (y0 , x0 ) .
∗
(2 + α )
for all t ∈ [0, 1].
(d (x1 , x2 ) + d (y1 , y2 ) ).
yn = G (yn−1 , xn−1 ),
0
1
y0 (t ) ≥ T3 y0 (t ) − T4 x0 (t ),
L
xn = G (xn−1 , yn−1 ),
(t − τ )α −1 x ( τ )d τ (α )
t
and
1 1 1 d (x1 , x2 ) + 2 d (y1 , y2 ) 2 2 2
x0 (t ) ≤ T3 x0 (t ) − T4 y0 (t ),
d (x , x ) (β − α + 1 ) 1 2 (2 − α ) L + d (y , y ) (2 + α ) (β − α + 1 ) 1 2
(t − τ )α +β −1 f (τ , x(τ )) dτ − λ (α + β )
and
(2 − α ) 1 − α 1 + α α + max | + t 0≤t≤1 (2 + α ) 4 2
≤
0
+
L d ( x1 , x2 ) 4(α + 1 )(β + 1 ) L + d ( y1 , y2 ) 2(α + 1 )(β + 1 )
=
t
(1 − τ )β −1 + f (τ , x(τ )) dτ 4(α + 1 ) 0 (β ) (2 − α ) 1 − α 1 + α 1 (1 − τ )β −α −1
+
×
1
(β − α ) L |λ| ≤ d (x , x ) + d (y , y ) (α + β + 1 ) 1 2 (α + 1 ) 1 2 L |λ| + d ( y1 , y2 ) + d ( x1 , x2 ) 2(α + β + 1 ) 2(α + 1 ) 0
335
2L 2|λ| L + + (α + β + 1 ) (α + 1 ) 2(α + 1 )(β + 1 ) 3 + α (2 − α ) 2L + , 4 (2 + α ) (β − α + 1 ) |λ| L L 2 = + + (α + 1 ) (α + β + 1 ) (α + 1 )(β + 1 ) (2 − α ) 2L + , (2 + α ) (β − α + 1 )
1 =
then it has a unique solution in C . Proof. It suffices to define G : C × C → C by G (x, y )(t ) = T3 x(t ) − T4 y(t ), and to apply Theorem 3.2.
the
same
arguments
as
in
the
proof
Example 3.1. Let us consider the following problem
3 1 1 D 2 D 2 + 14 x(t ) = 10 (x(t ) cos t ) − 1, 0 < t < 1, 1
1
x ( 0 ) + x ( 1 ) = 0, D 2 x ( 0 ) + D 2 x ( 1 ) = 0, D
21 2
x (0 ) + D
21 2
x ( 1 ) = 0.
of
(3.4)
336
H. Fazli, J.J. Nieto / Chaos, Solitons and Fractals 114 (2018) 332–337
1 1 3 1 10 (x (t ) cos t ) − 1, α = 2 , β = 2 and λ = 4 . Observe that, for every x ≥ y, we have 0 ≤ f (t, x ) − f (t, y ) ≤ 1 1 10 (x − y ) with L = 10 and
Here f (t, x ) =
2L |λ| L + + (α + β + 1 ) (α + 1 ) 2(α + 1 )(β + 1 ) 3 + α (2 − α ) 2L + = 0.42644, 4 (2 + α ) (β − α + 1 ) 2|λ| L L 2 = + + (α + 1 ) (α + β + 1 ) (α + 1 )(β + 1 ) (2 − α ) 2L + = 0.65289, (2 + α ) (β − α + 1 )
1 =
and = max{1 , 2 } = 0.65289 < 1. Now from (3.3) we let G : C × C → C by G (x, y )(t ) =
t
(t − τ )α+β −1 1
(x(τ ) cos τ ) − 1 (α + β ) 10 t (t − τ )α−1 ×dτ − λ y ( τ )d τ (α ) 0 1 1 (1 − τ )α+β −1 1 − (y(τ ) cos τ ) − 1 dτ 2 0 (α + β ) 10 1 λ (1 − τ )α−1 + x ( τ )d τ 2 0 (α ) 1 1 (1 − τ )β −1 1 + (x(τ ) cos τ ) − 1 dτ 4(α + 1 ) 0 (β ) 10 1 α β −1 t (1 − τ ) 1 − (y(τ ) cos τ ) − 1 2(α + 1 ) 0 (β ) 10 (2 − α ) 1 − α 1 + α α 1 (1 − τ )β −α−1 + + t (2 + α ) 4 2 (β − α ) 0 1 × (x(τ ) cos τ ) − 1 dτ
fractional orders in different intervals. As a first step, we establish the equivalence of this problem and the corresponding integral equation by applying the tools of fractional calculus and define a fixed point problem. The existence and uniqueness of solutions of these equations are obtained using Coupled fixed point theorems for mixed monotone mappings. Our approach is simple and is applicable to a variety of real world problems. For the illustration of the results, we have considered an example. As a special case, the existence results for a nonlinear thirdorder differential equations with anti-periodic boundary conditions:
D3 + λD2 x(t ) = f (t , x(t ) ),
x(0 ) + x(1 ) = 0, x (0 ) + x (1 ) = 0, x (0 ) + x (1 ) = 0, can be obtained by fixing α = 1 and β = 2 in the results of this paper.
0
10
(2 − α ) α+1 1 (1 − τ )β −α−1 − t (2 + α ) (β − α ) 0 1 × (y(τ ) cos τ ) − 1 dτ . 10
A relatively simple calculus, with the help of Maple, shows that (x0 (t ), y0 (t )) = (−1, 1 ) is a coupled lower and upper solution of problem (3.4). Therefore, all the assumption of Theorem 3.2 hold and consequently, problem (3.4) has a unique solution in C[0, 1]. Moreover, the unique solution of n n (3.4) can be obtained as lim n→∞ G (x0 , y0 ) where G (x0 , y0 ) = G G n−1 (x0 , y0 ), G n−1 (y0 , x0 ) . Using simple calculus, with the help of Maple, we have G (x0 , y0 ) = −0.33544 − 0.50 0 0 0t 2 + 0.10 0 0 0 cos(t )
√ 3 −0.43755 t + 0.61057t 2 ,
G (y0 , x0 ) = 0.24440 − 0.50 0 0 0t 2 − 0.10 0 0 0 cos(t )
√ 3 +0.28631 t + 0.72274t 2 .
It is interesting to point out that (G n (x0 , y0 ), G n (y0 , x0 ) ) n = 1, 2, 3, · · · serve as an approximation to the unique coupled fixed point of G of increasing accuracy as n → ∞. On the other hand, from Theorem 2.3, the unique solution of (3.4) can be obtained as
x∗ = lim G n (x0 , y0 ) = lim G n (y0 , x0 ). n→∞
n→∞
4. Conclusions In this paper, we have presented some results dealing with the existence and uniqueness of solutions for an anti-periodic boundary value problem of nonlinear Langevin equation involving two
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